SCMA012 Commercial Mathematics
DEPARTMENT OF BUSINESS
MANAGEMENT
SCMA012
COMMERCIAL MATHEMATICS
(BUSINESS STATISTICS)
1 | 2023 U n i v e r s i t y of Limpopo
,SCMA012 Commercial Mathematics
Table of Contents
Study Unit Title of section Page
1 Introduction to Estimation 2
1.1 The Estimation of Population Mean
1.2 The Estimation of Population Proportion
2 Introduction to Hypothesis Tests for Single Population 18
2.1 Hypothesis Test for One Population Mean
2.2 Hypothesis Test for One Population Proportion
3 Introduction to Hypothesis Tests for Two Populations 40
3.1 Hypothesis Test for Two Population Means
3.2 Hypothesis Test for Two Population Proportions
4 Chi-Square Distribution 54
4.1 Test of Independence
5 Simple Linear Regression 60
5.1 Regression Analysis
5.2 Correlation Analysis
Statistical Tables 71-76
References 77
Test Dates
1. Test 1 01 August 2023 09:00 -11:00
Scope: Unit 1
2. Test 2 04 September 2023 09:00 -11:00
Scope: Units 2 and 3
3. Test 3 09 October 2023 09:00 -11:00
Scope: Units 4 and 5
Prescribed textbook:
“Managerial Statistics, 9th Edition by Gerald Keller
Answers
“Answers to selected Units Exercises are provided”. All answers have been double-checked for
accuracy. However, I cannot be absolutely certain that there are no errors. Students should not
automatically assume that answers that don’t match provided answers are wrong. If you find any
errors, please you are welcomed to contact me and I will appreciate that.
2 | 2023 U n i v e r s i t y of Limpopo
,SCMA012 Commercial Mathematics
STUDY UNIT 1
INTRODUCTION TO ESTIMATION
1.1 INTRODUCTION
Having discussed descriptive statistics, probability distributions, and sampling distributions, in
SCMA011, first semester, we are now ready to discuss statistical inference. As we explained in
SCMA011, statistical inference is the process by which we acquire information and draw
conclusion about populations from samples (see Fig 1.1). Traditionally, there are two general
procedures for making inferences about population: Estimation and Hypothesis testing. With
Estimation procedures, we determine the approximate values of population parameters on the
basis of a sample statistics, and while with hypothesis testing, we must accept or reject assertions
(claims) about populations and/or their parameters.
In this unit, we introduce the concepts and foundations of estimation and demonstrate them with
simple examples.
Figure 1.1: Statistical Inference
1.2 CONCEPTS OF ESTIMATION
As its name suggests, the objective of estimation is to determine the approximate value of a
population parameter on the basis of a sample statistic. For example, the sample mean is
employed to estimate the population mean. We refer to the sample mean as the estimator of the
population mean. Once the sample mean has been computed, its value is called the estimate.
Problems of estimation can be found everywhere: in business, in science, as well as in everyday
life. In business,
(i) A chamber of commerce may want to know the average income of the families in its
community,
(ii) A real estate developer to know how many cars can be expected to drive by a certain
location per day
These examples, somebody is interested in determining the true value of some quantity, so they
are all problems of estimation. They would have been hypothesis testing, however, if the
chamber of commerce had wanted to decide on the basis of sample data whether the average
3 | 2023 U n i v e r s i t y of Limpopo
, SCMA012 Commercial Mathematics
family income in its community is really R43,000. Now it must be decided whether to accept or
reject a hypothesis (namely, an assertion or claim) about the parameter of a population.
1.2.1 Point Estimator and Point Estimate
1. A point estimator of a population parameter is a sample statistic used to estimate the
parameter. It draws inferences about a population by estimating the value of unknown
parameter using a single value or point.
2. A point estimate of a population parameter is the value of a sample statistic used to estimate
the parameter.
The most widely used point estimate of a population mean is the mean of a suitable sample. One
way to obtain information about a population mean 𝜇 without taking a census is to estimate it by
a sample mean 𝑥̅ , as illustrated in the next example.
Example 1.1: Estimating the Population Mean
Let us refer to a study in which industrial designers want to determine the average (mean) time it
takes an adult to assemble an “easy to assemble” toy. Using a random sample, they obtain the
following data (in minutes) for 36 persons who assembled the toy:
17 13 18 19 17 21 29 22 16 28
21 15 26 23 24 20 8 17 17 21
32 18 25 22 16 10 20 22 19 14
30 22 12 24 28 11
The mean of this sample is 𝑥̅ = 19.9 minutes, and this figure can be used as an estimate of , the
true average time it takes an adult to assemble the given toy. This kind of estimation is called a
point estimate, since it consists of single number, or a single point on the real number scale.
The point estimate of is 19.9
Activity 1.1 Overview Study Skill
Draw a mind - map of the different section or headings you will deal with in this study session.
Then go through the unit with the purpose of the completing the following map.
Confidence Interval Estimate
Confidence Interval
Confidence Interval Estimate
Estimate for the Mean
for the proportion
Confidence Interval Confidence Interval Estimate
Estimate when 𝜎 is known when 𝜎 is unknown
4 | 2023 U n i v e r s i t y of Limpopo
DEPARTMENT OF BUSINESS
MANAGEMENT
SCMA012
COMMERCIAL MATHEMATICS
(BUSINESS STATISTICS)
1 | 2023 U n i v e r s i t y of Limpopo
,SCMA012 Commercial Mathematics
Table of Contents
Study Unit Title of section Page
1 Introduction to Estimation 2
1.1 The Estimation of Population Mean
1.2 The Estimation of Population Proportion
2 Introduction to Hypothesis Tests for Single Population 18
2.1 Hypothesis Test for One Population Mean
2.2 Hypothesis Test for One Population Proportion
3 Introduction to Hypothesis Tests for Two Populations 40
3.1 Hypothesis Test for Two Population Means
3.2 Hypothesis Test for Two Population Proportions
4 Chi-Square Distribution 54
4.1 Test of Independence
5 Simple Linear Regression 60
5.1 Regression Analysis
5.2 Correlation Analysis
Statistical Tables 71-76
References 77
Test Dates
1. Test 1 01 August 2023 09:00 -11:00
Scope: Unit 1
2. Test 2 04 September 2023 09:00 -11:00
Scope: Units 2 and 3
3. Test 3 09 October 2023 09:00 -11:00
Scope: Units 4 and 5
Prescribed textbook:
“Managerial Statistics, 9th Edition by Gerald Keller
Answers
“Answers to selected Units Exercises are provided”. All answers have been double-checked for
accuracy. However, I cannot be absolutely certain that there are no errors. Students should not
automatically assume that answers that don’t match provided answers are wrong. If you find any
errors, please you are welcomed to contact me and I will appreciate that.
2 | 2023 U n i v e r s i t y of Limpopo
,SCMA012 Commercial Mathematics
STUDY UNIT 1
INTRODUCTION TO ESTIMATION
1.1 INTRODUCTION
Having discussed descriptive statistics, probability distributions, and sampling distributions, in
SCMA011, first semester, we are now ready to discuss statistical inference. As we explained in
SCMA011, statistical inference is the process by which we acquire information and draw
conclusion about populations from samples (see Fig 1.1). Traditionally, there are two general
procedures for making inferences about population: Estimation and Hypothesis testing. With
Estimation procedures, we determine the approximate values of population parameters on the
basis of a sample statistics, and while with hypothesis testing, we must accept or reject assertions
(claims) about populations and/or their parameters.
In this unit, we introduce the concepts and foundations of estimation and demonstrate them with
simple examples.
Figure 1.1: Statistical Inference
1.2 CONCEPTS OF ESTIMATION
As its name suggests, the objective of estimation is to determine the approximate value of a
population parameter on the basis of a sample statistic. For example, the sample mean is
employed to estimate the population mean. We refer to the sample mean as the estimator of the
population mean. Once the sample mean has been computed, its value is called the estimate.
Problems of estimation can be found everywhere: in business, in science, as well as in everyday
life. In business,
(i) A chamber of commerce may want to know the average income of the families in its
community,
(ii) A real estate developer to know how many cars can be expected to drive by a certain
location per day
These examples, somebody is interested in determining the true value of some quantity, so they
are all problems of estimation. They would have been hypothesis testing, however, if the
chamber of commerce had wanted to decide on the basis of sample data whether the average
3 | 2023 U n i v e r s i t y of Limpopo
, SCMA012 Commercial Mathematics
family income in its community is really R43,000. Now it must be decided whether to accept or
reject a hypothesis (namely, an assertion or claim) about the parameter of a population.
1.2.1 Point Estimator and Point Estimate
1. A point estimator of a population parameter is a sample statistic used to estimate the
parameter. It draws inferences about a population by estimating the value of unknown
parameter using a single value or point.
2. A point estimate of a population parameter is the value of a sample statistic used to estimate
the parameter.
The most widely used point estimate of a population mean is the mean of a suitable sample. One
way to obtain information about a population mean 𝜇 without taking a census is to estimate it by
a sample mean 𝑥̅ , as illustrated in the next example.
Example 1.1: Estimating the Population Mean
Let us refer to a study in which industrial designers want to determine the average (mean) time it
takes an adult to assemble an “easy to assemble” toy. Using a random sample, they obtain the
following data (in minutes) for 36 persons who assembled the toy:
17 13 18 19 17 21 29 22 16 28
21 15 26 23 24 20 8 17 17 21
32 18 25 22 16 10 20 22 19 14
30 22 12 24 28 11
The mean of this sample is 𝑥̅ = 19.9 minutes, and this figure can be used as an estimate of , the
true average time it takes an adult to assemble the given toy. This kind of estimation is called a
point estimate, since it consists of single number, or a single point on the real number scale.
The point estimate of is 19.9
Activity 1.1 Overview Study Skill
Draw a mind - map of the different section or headings you will deal with in this study session.
Then go through the unit with the purpose of the completing the following map.
Confidence Interval Estimate
Confidence Interval
Confidence Interval Estimate
Estimate for the Mean
for the proportion
Confidence Interval Confidence Interval Estimate
Estimate when 𝜎 is known when 𝜎 is unknown
4 | 2023 U n i v e r s i t y of Limpopo
